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I will report on joint work with Ben Howard, Andrew Snowden and Ravi Vakil on finding a presentation for the ring of polynomial invariants (under projective transformations) of n ordered points on the projective line. This problem was much studied in the 19-th century. In particular Kempe proved that the ring of invariants was generated by the invariants of lowest degree (an exceptional phenomenon not true for the case of unordered points on the line or ordered points in the projective plane). I will first present our new proof of Kempe's theorem. I will sketch this proof to give you an idea of the connection of my talk with combinatorics.

The first step is classical; one interprets the invariant monomials in terms of regular graphs on 1,2,.., n. Every invariant is the sum of invariant monomials. Multiplying monomials is just superimposing one graph on another. Another key classical step involves interpreting the Plucker relation among invariants graphically (as an "uncrossing relation" for edges of the graph)

       _
x = _ + | |

The degree of an invariant is the valence of the regular graph. There is a simple reduction to the case in which the number of points is even. One then uses Plucker to replace the graph by a regular bipartite graph G. Then comes the punch line: apply the Hall marriage theorem to deduce that one can factor off a matching from G. To summarize, Hall -> Kempe.

After that I will explain the "simple binomial quadratic relations" among the invariants. We conjecture these generate the ideal of all relations. We prove they radically generate in the sense that the radical of the ideal they generate is the ideal of all relations. So the simple binomial relations cut out the corresponding projective variety (the Kempe generators give a projective embedding of the moduli space).

We have found generators for the full ideal of relations using a certain toric degeneration of the above ring. These generators are of degree two, three and four so our presentation is more complicated than we would like. However there are many toric degenerations. There is one degeneration for each trivalent tree with n leaves or each triangulation of a regular n-gon or each topological class of pair of pants decompositions of the n holed two-sphere. One uses the tree to weight the above monomials thereby filtering the ring. By standard theory one has a flat degeneration to the associated graded ring which is toric. The degeneration at the beginning of this paragraph came from the triangulation where all diagonals emanate from a fixed vertex. The second description and this particular triangulation connects up with work I did with Misha Kapovich on bending flows of polygonal linkages in R^3 in 1995. However Andrew, Ben, Ravi and I have recently discovered that there are other triangulations that are better behaved and it appears possible that we may be able to prove our conjecture using one of these better behaved toric degenerations.